Blake Keeler, UNC-Chapel Hill – GMA & Visions Seminar

Title: Diagonalizing the Laplacian – A Tourist’s Guide to Spectral Theory

Abstract: This talk is aimed at first and second year grad students who are interested in analysis, or really anyone with a passing curiosity about what analysts do all day besides fiddle with epsilons and deltas. It will be a (hopefully) gentle introduction to some basic notions that we consider in the spectral theory of operators. Spectral theory is motivated by what we know about eigenvalues and eigenvectors for finite dimensional operators, i.e. matrices. We will be exploring to what extent some of the properties of eigenvalues and eigenvectors of matrices carry over to operators on infinite dimensional spaces. In particular, we will examine the notion of diagonalizing an operator in a “basis of eigenvectors,” whatever that means for an infinite dimensional object. In fact, we will focus on the very special operator known as the Laplacian. There is a very concrete and straightforward way to understand the concept of diagonalization for this operator acting on functions on a compact domain, but it will be much more subtle if we consider the Laplacian on all of R^n. My goal will be to highlight both the differences and similarities in the spectral properties of the Laplacian in these two cases.